73
votes
Accepted
Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?
First, there is indeed nothing mathematically very deep in this observation, and I agree that the word "breakthrough" might be exaggerated. But on the other hand lots of very deep ideas look trivial ...
- 42.4k
47
votes
Books/websites which have motivating stories of mathematicians overcoming hardships in life
The AMS published a book called Living Proof in which a number of mathematicians relate their own experience with overcoming adversity. Some of these are famous although most are ordinary ...
44
votes
Why is the definition of the higher homotopy groups the "right one"?
I think that obstruction theory is one of the most important reasons to study homotopy groups. If you are interested in studying the possible homotopy classes of maps $X \to Y$ of spaces where $X$ has ...
43
votes
Accepted
Why did Euler consider the zeta function?
This history is described in Euler and the Zeta Function by Raymond Ayoub (1974). In his early twenties, around 1730, Euler considered the celebrated problem to calculate the sum $$\zeta(2)=\sum_{n=1}^...
- 188k
41
votes
Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?
The other answers are quite good and nothing is wrong with them, but lest a wrong impression be given (e.g. by the implicit suggestion that the idea of "categories as sets in the next dimension" in ...
- 66.8k
38
votes
Why study finite topological spaces?
Long comment:
Note that a finite topological space is nothing but a finite pre-order (order, iff the topology is $T_0$), given by $x\leq y$ iff $x\in \overline{\{y\}}$. An equivalence relation on a ...
- 11.6k
33
votes
Why to believe the Fargues geometrization conjecture?
We finally have finished our paper, detailing the conjecture! We have also included an extensive introduction that I hope gives some impression of why one might hope for such a statement, and I'll ...
- 21.3k
30
votes
Why is the definition of the higher homotopy groups the "right one"?
The calculation of the ring of manifolds up to cobordism was made possible thanks to identifying this ring with the homotopy groups of certain Thom spaces, and then applying methods of homotopy theory....
30
votes
Why to believe the Fargues geometrization conjecture?
Why believe in it ? Because I said so. There is absolutely no doubt it is true.
This is in some sense evident to me now, even more than in 2014 after my talk at the MSRI, in particular after the ...
- 417
29
votes
Why is the definition of the higher homotopy groups the "right one"?
I think there's something fundamental missing from all the other answers so far: the modern realization that topological spaces are distinct from $\infty$-groupoids.
Suppose you didn't know about ...
28
votes
Why study finite topological spaces?
As Uri Bader's answer notes, finite $T_0$ topological spaces are equivalent to finite partially ordered sets (posets). Now, combinatorialists who are interested in the topology of finite posets most ...
- 82.7k
25
votes
Why to believe the Fargues geometrization conjecture?
These notes, from a course Fargues taught at Chicago and transcribed by Sean Howe, are very nice and make a very strong effort to motivate this conjecture and the surrounding theory by analogy with '...
- 3,058
24
votes
Accepted
Why study finite topological spaces?
Prompted by the same excerpt, I asked Bill Thurston in 2011:
Can you point me somewhere
where I can read about some of your mental models and structures as
they relate to finite topological spaces?
...
- 4,994
22
votes
Accepted
Entering to the K-theory realm
I think that doing algebraic K-theory properly certainly requires a good background on stable homotopy theory, that is to say the homotopy theory of spectra. Unfortunately there are not many textbooks ...
- 16.5k
20
votes
Why is the definition of the higher homotopy groups the "right one"?
There are many things to say here. Here's one. Suppose you want to classify all spaces up to (weak) homotopy equivalence, or equivalently all CW complexes up to homotopy equivalence. The zeroth step ...
19
votes
Books/websites which have motivating stories of mathematicians overcoming hardships in life
A good overview (including several mathematicians with an unusual background) is compiled by Nassim Nicholas Taleb at Bookauthority. Even without buying the books, the list may give you pointers to ...
19
votes
Why study finite topological spaces?
Any finite simplicial complex is weakly homotopy equivalent to a finite topological space and vice versa:
Are finite spaces a model for finite CW-complexes?
So, when it comes to computing homotopy ...
- 2,934
18
votes
Accepted
Ultraproducts of Banach spaces versus model theoretic ultraproduct
The ultraproduct of Banach spaces is the ultraproduct in the sense of metric structures in continuous logic. For a nice survey on this topic, see Model theory for metric structures by Ben Yaacov, ...
- 5,031
16
votes
What motivations for automorphic forms?
To give a brief answer, which I think applies to all audiences, and I hope is not too "elementary" for you (I'm not attempting to give details, which of course need to be specialized for the intended ...
16
votes
Why is the definition of the higher homotopy groups the "right one"?
Many answers are going to seem at least a little circular, including this one. If you agree that CW complexes are an interesting class of spaces, encompassing any spaces you may want to consider, then ...
16
votes
Ultraproducts of Banach spaces versus model theoretic ultraproduct
As a logician, I take the model-theoretic notion of ultraproduct as the primary one, so the following formal connection describes how to get the Banach-space ultraproduct from the model-theoretic one.
...
- 73.2k
15
votes
Why is the definition of the higher homotopy groups the "right one"?
An axiomatic characterization of homotopy groups similar to that for (co)homology theories is given in Section 4 of Milnor's classic paper
Construction of Universal Bundles, I
John Milnor
...
15
votes
Accepted
What motivations for automorphic forms?
The specific issue of what automorphic forms on bigger groups than $GL(2)$ over $\mathbb Q$ (for example) may tell us about automorphic forms (and L-functions) for $GL(2,\mathbb Q)$ or $GL(1,k)$ for ...
15
votes
Accepted
Motivation for relative schemes: why should one work with schemes over a ringed topos?
The comments to your question discuss the variation of relative schemes over a topos, vs relative schemes over a site. But it seems your question
stood at the more basic level of the relevance of ...
- 12.9k
14
votes
Accepted
What is the significance of Friedlander-Iwaniec and related theorems?
I think questions about prime values of polynomials are considered inherently interesting.
All these questions are special cases of Bunyakovsky's conjecture, or, if you want, the Bateman-Horn ...
- 148k
14
votes
Books/websites which have motivating stories of mathematicians overcoming hardships in life
The "Unreal Life of Oscar Zariski" by Carole Parikh is very inspiring.
14
votes
Why study finite topological spaces?
And what would be some instances where 'standard circumlocutions' used to avoid them?
For example, one can define contractability and compactness (for metrizable spaces) in terms of maps of finite ...
- 141
13
votes
Books/websites which have motivating stories of mathematicians overcoming hardships in life
Alexander Grothendieck was born the son of Russian anarchist Jew in Nazi Germany, was imprisoned in a concentration camp as a young stateless child, had to hide from the Nazis, lived as a miserable ...
12
votes
What motivations for automorphic forms?
Here's an answer for audiences (C), (D) about class field theory:
Class field theory provides a way classify abelian extensions of number fields $F$. In the case of $F=\mathbb Q$, this is answered ...
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